Lecture 5.1: The Rhythm of Nature

A wave is a transfer of energy, not matter.

Today’s Essential Questions

• What is a wave, and what does it transfer?
• What is the difference between a transverse and a longitudinal wave?
• What is the relationship between wave speed, frequency, and wavelength?

Connecting to Our Last Investigation

In the Slinky lab and simulations, you created disturbances and watched them travel. You saw two distinct types of waves. Today we will give those waves formal names—transverse and longitudinal—and learn the mathematical language that describes them all.

What is a Wave?

A wave is a disturbance that propagates, or travels, through a medium.

The most crucial idea is that waves transfer energy from one place to another, but they do not transfer matter. The particles of the medium oscillate around a fixed equilibrium position, but they don’t travel with the wave.

Think of a stadium wave: people stand up and sit down, but they don’t run around the stadium. The energy of the cheer travels.

Two Types of Mechanical Waves

1. Transverse Waves: The particles of the medium oscillate perpendicular (at a right angle) to the direction of the wave’s motion.
   • Examples: A wave on a guitar string, ripples on a pond, and all electromagnetic waves (like light).
2. Longitudinal Waves: The particles of the medium oscillate parallel to the direction of the wave’s motion.
   • Examples: Sound waves, a compressed slinky. These waves are made of compressions and rarefactions.

The Anatomy of a Wave

We use a few key terms to describe the “structure” of a wave:

• Amplitude: The maximum displacement from the equilibrium position. It’s related to the wave’s energy.
• Wavelength ($\lambda$): The length of one complete wave cycle (e.g., from crest to crest).
• Frequency (f): The number of complete cycles that pass a point per second. Measured in Hertz (Hz).
• Period (T): The time it takes for one complete cycle to pass. It is the inverse of frequency ($T = 1/f$).

The Wave Speed Equation

There is a simple, powerful equation that connects three of these properties. The speed of a wave is its frequency multiplied by its wavelength.

\(v = f \lambda\) \(\text{Wave Speed} = \text{Frequency} \times \text{Wavelength}\)

This makes intuitive sense: a wave’s speed depends on how many waves pass per second (frequency) and how long each of those waves is (wavelength).

Wave Speed by the Numbers

A sound wave has a frequency of 440 Hz (the musical note A) and a wavelength of 0.78 meters. How fast is it traveling through the air?

\(v = f \lambda\) \(v = (440 \text{ Hz})(0.78 \text{ m})\) \(v = 343.2 \text{ m/s}\)

This is the speed of sound in air at room temperature.

Thinking Lens: Structure and Function

The ‘structure’ of a wave is defined by its properties like frequency, wavelength, and amplitude.

Question: How does changing a wave’s structure affect its ‘function’—the way it transfers energy and interacts with the world? For example, how does the function of a high-frequency sound wave (a high-pitched note) differ from that of a low-frequency sound wave (a low-pitched note)?

Preparing for Our Next Task

The wave speed equation, v = fλ, is the central tool you will use to solve every problem in the upcoming Problem Set 1 (Frequency & Wavelength). You will use it to calculate speed, frequency, or wavelength when given the other two variables.

Summary: Answering Our Questions

• What is a wave, and what does it transfer? A wave is a disturbance that travels through a medium, transferring energy without transferring matter.

• What is the difference between transverse and longitudinal waves? In a transverse wave, particles move perpendicular to the wave’s direction; in a longitudinal wave, they move parallel to it.

• What is the relationship between wave speed, frequency, and wavelength? They are related by the wave speed equation, $v = f \lambda$.

Prompt: In 2-3 sentences, define a wave and explain the key properties that are used to describe it.